# How do you simplify 5(35(3^n-1) - 29(2^n-1) ) - 6(35(3^n-2) - 29(2^n-2) )?

Oct 2, 2015

In the unlikely event that I didn't make an arithmetic mistake:
$\textcolor{w h i t e}{\text{XXXX}} - 35 \left({3}^{n}\right) + 275 \left({2}^{n}\right) + 42$

#### Explanation:

Note: This isn't really difficult; it's just tedious.

5(35(3^n−1)−29(2^n−1))−6(35(3^n−2)−29(2^n−2))

$= \left(5 \times 35\right) \left({3}^{n} - 1\right) - \left(5 \times 29\right) \left({2}^{n} - 1\right) + \left(\left(- 6\right) \times 35\right) \left({3}^{n} - 2\right) - \left(- 6 \times 29\right) \left({2}^{n} - 2\right)$

$= 175 \left({3}^{n} - 1\right) - 145 \left({2}^{n} - 1\right) + \left(- 210\right) \left({3}^{n} - 2\right) + 174 \left({2}^{n} - 2\right)$

$= 175 \cdot \textcolor{red}{{3}^{n}} \textcolor{g r e e n}{- 175} - 145 \cdot \textcolor{b l u e}{{2}^{n}} \textcolor{g r e e n}{+ 145} - 210 \cdot \textcolor{red}{{3}^{n}} \textcolor{g r e e n}{+ 420} + 174 \cdot \textcolor{b l u e}{{2}^{n}} \textcolor{g r e e n}{- 348}$

$= \left(175 - 210\right) \cdot \textcolor{red}{{3}^{n}} + \left(- 145 + 420\right) \cdot \textcolor{b l u e}{{2}^{n}} + \left(\textcolor{g r e e n}{- 175 + 145 + 420 - 348}\right)$

$= - 35 \left({3}^{n}\right) + 275 \left({2}^{n}\right) + 42$